\documentstyle[titlepage,leqno,11pt]{article} % equation numbering section.number \makeatletter \@addtoreset{equation}{section} \def\theequation{\thesection.\arabic{equation}} \makeatother \makeatletter \newcommand{\no}{\noindent} \newcommand{\Om}{\Omega} %\newcommand{\r}{{\bf R}} \newcommand{\eps}{\varepsilon} \newcommand{\rn}[1]{{\bf R}^{#1}} \newcommand{\hh}{{\cal H}^{m}} \newcommand{\h}[2]{{\cal H}^{#1}\bigl({#2}\bigr)} \newcommand{\dis}[2]{{\rm dist}^2({#2},{#1})} \newcommand{\E}[2]{\int_{#2}|\nabla {#1}|^2\,dx} \newcommand{\F}[2]{\int_{{#2}\cap S_{#1}}|T^\perp x|^2\,d{\cal H}^m} \newcommand{\dc}{D_{\hbox{\rm c}}} \newcommand{\dd}{\bar{D}_{\hbox{\rm c}}} \newcommand{\mm}[1]{{\cal M}_\omega(#1)} \newcommand{\dev}[2]{{\rm Dev}(#1,#2)} \newcommand{\res}{\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits} \newcommand{\qed}{\thinspace\null\nobreak\hfill\hbox{\vbox{\kern- .2pt\hrule height.2pt depth.2pt\kern-.2pt\kern-.2pt \hbox to2.5mm{\kern-.2pt\vrule width.4pt \kern-.2pt\raise2.5mm\vbox to.2pt{}\lower0pt\vtop to.2pt{}\hfil\kern- .2pt \vrule width.4pt\kern-.2pt}\kern-.2pt\kern-.2pt\hrule height.2pt depth.2pt \kern-.2pt}}\par\medbreak} \newtheorem{definition}{Definition}[section] \newtheorem{lemma}[definition]{Lemma} \newtheorem{theorem}[definition]{Theorem} \newtheorem{proposition}[definition]{Proposition} \newtheorem{corollary}[definition]{Corollary} \newtheorem{remark}[definition]{Remark} \title{Higher regularity of solutions \\ of free discontinuity problems} %\title{} \author{} \author{{\Large Luigi Ambrosio} \\ Dipartimento di Matematica \\ ``F. Casorati'', Via Abbiategrasso 205 \\27100 Pavia, Italy \and {\Large Nicola Fusco} \\ Dipartimento di Matematica \\ ``U. Dini'', Viale Morgagni 67/A\\ 50134 Firenze, Italy \and {\Large Diego Pallara} \\ Dip. di Matematica \\ Universit\`a di Lecce \\ C.P. 193, 73100 Lecce, Italy} \begin{document} \maketitle \begin{abstract} In this paper we continue the analysis, started in \cite{ap}, \cite{afp} of the regularity of solutions of free discontinuity problems. We choose as a model problem the minimization of the Mumford-Shah functional. Assuming that in some region the optimal discontinuity set $\Gamma$ is the graph of a $C^{1,\rho}$ function, we look for conditions ensuring the higher regularity of $\Gamma$. Our results are optimal in the two dimensional case. As an application, we prove that in the case of the Mumford-Shah functional and in similar problems the Lavrentiev phenomenon does not occur. \end{abstract} \hbox{}\vfill \tableofcontents \vfill\eject \section{Introduction} Free discontinuity problems are variational problems where the functional to be minimized depends on a pair $(u,K)$, where $K$ is a hypersurface, or more generally a closed set, and $u$ is usually smooth outside $K$. The model problem is the Mumford-Shah functional, proposed in \cite{mumsh} as a variational model of image segmentation: \begin{equation} \label{msfunct} F(u,K)=\int_{\Omega\setminus K} \bigl[ |\nabla u|^2 +\alpha|u-g|^2 \bigr]\, dx + \beta{\cal H}^{n-1}(\Omega\cap K); \end{equation} here, $\Omega\subset\rn{n}$ is an open set, $\alpha,\,\beta>0$, $g\in L^\infty(\Omega) \cap L^2(\Omega)$ and ${\cal H}^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure. We refer to \cite{mumsh}, \cite{morsol} for a comprehensive discussion of this model. The existence of a minimizing pair $(u,K)$, with $K$ closed and $u\in C^1(\Omega\setminus K)$, has been proved by De Giorgi, Carriero and Leaci in \cite{dcl} (see also \cite{dms}) using the theory of $SBV$ functions introduced in \cite{dga}. But, while the smoothness of the function $u$ in $\Omega\setminus K$ can be easily investigated using classical arguments based on first variation and elliptic equations techniques, as regards the regularity of $K$ the existence theory ensures only that $K$ is rectifiable -- a very mild regularity property. Recently, we faced the problem of the partial regularity of the discontinuity set $K$: our main results are presented in \cite{ap}, \cite{afp} (see also \cite{ap2} for a brief survey) and are based on a double blow-up technique, suitable to exploit the interaction between the volume term and the surface term in functionals like $F$ in (\ref{msfunct}). Our results, which hold for a class of free discontinuity problems including the Mumford-Shah functional, can be briefly summarized by stating that if $(u,K)$ is a minimizing pair then $K$ is locally a $C^{1,\rho}$ hypersurface up to a ${\cal H}^{n-1}$ negligible singular set $S$. Some other results, valid only for the Mumford-Shah functional in two space dimensions, have been obtained with different methods in \cite{davsemm}, \cite{bon}. The aim of this paper is to improve the regularity results on the discontinuity set $K$: in particular, we focus our attention on the Mumford-Shah functional and show the following result, which is an immediate consequence of Theorem \ref{meanreg1}: \begin{theorem} Let $(u,K)$ be a minimizer of (\ref{msfunct}); then, up to a ${\cal H}^{n-1}$-negligible singular set $S$, the minimizing set $K$ is (locally) the graph of a $C^{1,1}$ function if the space dimension is 2, and of a $H^2$ function $\phi$ with gradient in $C^{0,\rho}$ for any $\rho<1$ if $n>2$. \end{theorem} Moreover (see Theorem \ref{meanreg2}), if the datum $g$ is assumed to belong to $C^{k,\rho}$ with $k\geq 1$ and $\rho\in (0,1]$, then (for any $n\geq 2$) $\phi\in C^{k+2,s}$ for some $s\in (0,1]$. The basic idea can be explained as follows: given a minimizing pair $(u,K)$, by the first variation formula, see \cite{clpp}, \cite{mumsh}, it is known that the function $u$ solves the equation $\Delta u=u-g$ in $\Omega\setminus K$ with homogeneous Neumann boundary conditions on $K$, and moreover, under regularity assumptions, the mean curvature of $K$ equals the jump of $|\nabla u|^2$ across it. It follows that $\phi$ is a weak solution of a mean curvature equation, and this can be exploited to get further regularity. The first step is the proof of the regularity of $\nabla u$ up to (both sides of) $K$: this amounts to prove a boundary regularity result for elliptic problems in $C^{1,\rho}$ domains which seems to be interesting by itself. This is achieved in \S 2, proving that $\nabla u$ has H\"older continuous traces on both sides of $K$. In \S 3 we prove a regularity result for weak solutions of mean curvature equations which will be used in \S 4 to prove the announced regularity theorem on the discontinuity set $K$. Finally, extending to any dimension the results of \cite{dise}, in \S 5 we use our regularity results to prove that in Mumford-Shah problem (and even in more general situations, see Remark~\ref{Lav} below) the so called Lavrentiev phenomenon does not occur, {\em i.e.} the minimum of $F$ among all the pairs $(u,K)$ with $K$ closed and $u\in C^1(\Omega\setminus K)$ agrees with the infimum of $F$ among all the pairs $(u,K)$ with $K$ piecewise smooth and $u\in C^\infty(\Omega\setminus K)$. This will easily follow by a general approximation result concerning $SBV$ functions. Finally, we remark that all the results of this paper are independent of \cite{ap}, \cite{afp}, and of the existence of minimizers; we simply assume that a minimizing pair $(u,K)$ exists and that, in some region $A$, $K$ is the graph of a $C^{1,\rho}$ function. It has been proved in \cite{ap}, \cite{afp} that ${\cal H}^{n-1}$-all points of a minimizer $K$ have a neighbourhood $A$ where this property holds. This paper, except for the last section, is independent also of the theory of $SBV$ functions, whose main properties will be recalled at the beginning of the last section. \section{Regularity of Neumann problems in $C^{1,\rho}$ domains} Let us recall the definition of the Morrey and Campanato spaces in a regular domain $\Om\subset\rn{n}$. For more information we refer to \cite{gia}. The space $L^{2,\lambda}(\Om)$, with $0\leq\lambda\leq n$, is the Morrey space of the $L^2(\Om)$ functions $u$ satisfying $$ \sup_{x\in\Omega,\,\varrho>0} \varrho^{-\lambda} \int_{B_\varrho(x)\cap\Omega}|u|^2\, dy<+\infty ; $$ for $0\leq\lambda\leq n+2$, ${\cal L}^{2,\lambda}(\Om)$ denotes the Campanato space of the $L^2(\Om)$ functions satisfying $$ \sup_{x\in\Omega,\,\varrho>0} \varrho^{-\lambda} \int_{B_\varrho(x)\cap\Omega}|u-u_{x,\varrho}|^2\, dy<+\infty, $$ where $u_{x,\varrho}$ is the mean value of $u$ in $B_\varrho(x)\cap\Omega$. Finally, notice that for $\lambda>n$, ${\cal L}^{2,\lambda}(\Om)$ is isomorphic to $C^{0,\gamma}(\overline{\Om})$ for $\gamma=(\lambda- n)/2$. Let $u\in H^1(\Om)$ be satisfying, in the distribution sense \begin{equation}\label{eq66} -\Delta u=f\quad\hbox{\rm in }\Om,\qquad\qquad \frac{\partial u}{\partial n}=0\quad\hbox{\rm on }\Gamma \end{equation} where $\Gamma\subset\partial\Om$ is a $C^{1,\rho}$ hypersurface, relatively open in $\partial\Om$ and $f\in L^\infty(\Om)$. In this section we investigate on the regularity of $\nabla u$ near $\Gamma$. In the two dimensional case the regularity of $\nabla u$ can be easily achieved using conformal mappings and Shwartz's reflection principle: \begin{theorem}\label{neureg1} Assume $n=2$. Then $\nabla u$ has an extension to $\Om\cup\Gamma$ which is locally $C^{0,s}$ for any $s<\rho$. \end{theorem} {\sc Proof.} It is not restrictive to assume $\rho<1$. For any $x_0\in\Gamma$ and for $\varrho>0$ sufficiently small there is a $C^{1,\rho}$ conformal mapping $\Phi$ between $B_\varrho(x_0)\cap\Om$ and a half ball $B^+$ such that $B_\varrho(x_0)\cap\partial\Om=B_\varrho(x_0)\cap\Gamma$ is mapped onto the flat part $H$ of $\partial B^+$. Setting $$v(y)=u(\Phi^{-1}(y)), \qquad g(y)=f(\Phi^{-1}(y))|{\rm det} J\Phi^{-1}(y)|$$ the function $v$ solves $\Delta v=g$ in $B^+$ with homogeneous Neumann boundary conditions on $H$. By the boundary regularity of conformal mappings between $C^{1,\rho}$ domains (see for instance \cite{po}, Theorem 3.6) $g$ is bounded near $H$. By an easy reflection argument the functions $v$ and $g$ can be extended to the whole ball $B$ in such a way that $\Delta v=g$ in $B$. In particular, $v\in C^{1,s}_{\rm loc}(B)$ for any $s<1$. Since $u=v\circ\Phi$, again the $C^{1,\rho}$ regularity of $\Phi$ implies our statement.\qed In higher dimensions, we still use a change of domain and a reflection argument. However, the change of variables changes the equation hence the proof is much more technical and our result is not sharp as in dimension 2. \begin{theorem}\label{neureg} Under the assumptions above, the function $\nabla u$ has a locally H\"older continuous extension to $\Om\cup\Gamma$. \end{theorem} Equations (\ref{eq66}) are equivalent to \begin{equation}\label{eq1} \int_\Om \langle\nabla u(x),\nabla\varphi(x)\rangle\,dx= \int_\Om f(x)\varphi(x)\,dx \end{equation} for any $\varphi\in C^1_0(\rn{n})$ with $\varphi=0$ on $\partial\Omega\setminus\Gamma$. We know that for any $x_0\in\Gamma$ there is a $C^{1,\rho}$ diffeomorphism $\Phi$ of a neighbourhood $U$ of $x_0$ and the unit cube $Q$ mapping $U\cap\Om$ into $Q^+=Q\cap\{x_n>0\}$ and $U\cap\partial\Om$ into $H=Q\cap\{x_n=0\}$. Then, assuming that $U$ is sufficiently small to have the inclusion $U\cap\partial\Om\subset\Gamma$, condition (\ref{eq1}) implies \begin{equation}\label{eq2} \int_{Q^+} A_{ij} \frac{\partial v}{\partial y_i} \frac{\partial\phi}{\partial y_j}\,dy=\int_{Q^+} g(y)\phi(y)\,dy \end{equation} for any $\phi\in C^1_0(\rn{n})$ with $\phi=0$ on $\partial Q^+\setminus H$, and with $v(y)=u\bigl(\Phi^{-1}(y)\bigr)$, $g(y)= f\bigl(\Phi^{-1}(y)\bigr)\bigl\vert{\rm det\,} J\Phi^{-1}(y) \bigr\vert$ and \begin{equation}\label{eq3} A_{ij}(y)=\sum_{k=1}^n \frac{\partial\Phi^i}{\partial x_k}\bigl(\Phi^{-1}(y)\bigr) \frac{\partial\Phi^j}{\partial x_k}\bigl(\Phi^{-1}(y)\bigr) \bigl\vert{\rm det\,} J\Phi^{-1}(y)\bigr\vert. \end{equation} The fuctions $A_{ij}$ are H\"older continuous and $g$ is bounded. Moreover, choosing $U$ sufficiently small and $\Phi$ sufficiently close in $C^1$ norm to a rotation we can assume that the matrix $A$ is uniformly elliptic, {\em i.e.}, \begin{equation}\label{eq4} A_{ij}(y)\xi_i\xi_j\geq \theta|\xi|^2\qquad\qquad\forall\xi\in\rn{n},\,\, \forall y\in Q \end{equation} for some constant $\theta>0$. Then, since $x_0\in\Gamma$ is arbitrary, Theorem~\ref{neureg} is a straightforward consequence of the following result: \begin{theorem}\label{neureg2} Let $v\in H^1(Q^+)$ be satisfying (\ref{eq2}) with the matrix $A$ $\rho$-H\"older continuous, uniformly elliptic in $Q^+$ and $g\in L^\infty(Q^+)$. Then, $\nabla v$ has a locally H\"older continuous extension to $Q^+\cup H$. \end{theorem} The proof of Theorem \ref{neureg2} will be achieved by a reflection argument. For any $x=(x',x_n)\in Q^-$ we define $v(x)=v(x',-x_n)$, $g(x)=g(x',-x_n)$ and $$ A_{ij}(x',x_n)=\cases{\,\,\,\,A_{ij}(x',-x_n) & if either $1\leq i,\,j0$; \cr A^- & if $x_n<0$} $$ and we denote by $w\in H^1(B_R)$ the solution of the problem \begin{equation}\label{zeq1} -{\rm div}\bigl(\bar{A}(x-x_0)\nabla w(x)\bigr)=0,\qquad\qquad w(x)=v(x+x_0)\quad\hbox{\rm on }\partial B_R. \end{equation} The H\"older continuity of $A$ implies \begin{equation}\label{zeq2} |A(x)-\bar{A}(x)|\leq cR^\rho\qquad\qquad\forall x\in B_R(x_0). \end{equation} Setting $$ \dd w(x)= \sum_{i=1}^n \bar{A}_{in}(x) \nabla_i w(x)$$ the plan of the proof is the following: in the first step we estimate $D_\tau w$; in the second step we estimate $\dd w$; in the last ones we get a $L^{2,\lambda}$ estimate on $\nabla v$ and eventually ${\cal L}^{2,\mu}$ estimates which lead to the H\"older continuity of $D_\tau w$ and $\dc w$. {\sc Step 1.} Using the fact that $\bar{A}$ is constant in the tangencial directions, a standard application of Nirenberg's difference quotient method gives that $D_\tau w\in H^1_{\rm loc}(B_R)$ and the Caccioppoli estimate \begin{equation}\label{zeq55} \int_{B_\varrho}|\nabla (D_\tau w)|^2\,dx\leq c\varrho^{-2} \int_{B_{2\varrho}}|D_\tau w-(D_\tau w)_{2\varrho}|^2\,dx \end{equation} for $0<\varrho\leq R/2$, where as usual $(h)_\varrho$ denotes the average of the function $h$ in $B_\varrho$. Moreover, $D_\tau w$ satisfies $$ \int_{B_R}\bar{A}_{ij} \frac{\partial D_\tau w}{\partial x_i} \frac{\partial \phi}{\partial x_j}\,dx=0\qquad\qquad\forall\phi\in C^1_0(B_R). $$ The regularity theory for solutions $u$ of elliptic equations with measurable coefficients (see for instance \cite{gia}, page 80 and page 82) yields the estimates $$\varrho^n\max_{B_\varrho}|u|^2\leq c\int_{B_{2\varrho}}|u^2(x)|\,dx \qquad 0<\varrho\leq R/2$$ $$\int_{B_\varrho}|\nabla u|^2\,dx\leq c\biggl(\frac{\varrho}{R}\biggr)^{n-2+2\gamma}R^{-2} \int_{B_R}|u-(u)_R|^2\,dx\qquad 0<\varrho\leq R.$$ \noindent Applying these estimates to $u=D_\tau w$ we infer \begin{equation}\label{zeq3} \varrho^n\sup_{B_\varrho} |D_\tau w|^2\leq c\int_{B_{2\varrho}} |D_\tau w|^2\,dx \qquad\qquad 0<\varrho\leq R/2 \end{equation} \begin{eqnarray} \label{zeq4} &&\int_{B_\varrho}|D_\tau w-(D_\tau w)_\varrho|^2\,dx \nonumber\\ &\leq& c\biggl(\frac{\varrho}{R}\biggr)^{n+2\gamma} \int_{B_R}|D_\tau w-(D_\tau w)_R|^2\,dx\qquad 0<\varrho\leq R \end{eqnarray} \noindent for some $\gamma>0$ depending only on $n$, the ellipticity constant $\theta$ and the $L^\infty$ norm of $A$. {\sc Step 2.} Since $D_\tau (\dd w)=\dd (D_\tau w)$, we have that $\dd w$ has tangencial distributional derivatives in $L^2_{\rm loc}(B_R)$. To estimate the normal derivative of $\dd w$ we use equation (\ref{zeq1}). Indeed, the equality $$ \nabla_n (\dd w)=-\sum_{i=1}^{n-1}\sum_{j=1}^n \bar{A}_{ij} \nabla^2_{ij} w $$ implies that also $\nabla_n (\dd w)$ belongs to $L^2_{\rm loc}(B_R)$. Moreover, we have the estimate \begin{equation}\label{zeq5} |\nabla (\dd w)|\leq c|\nabla (D_\tau w)|. \end{equation} Using Poincar\'e inequality, (\ref{zeq5}) and (\ref{zeq55}) we get \begin{eqnarray*} \int_{B_\varrho}|\dd w-(\dd w)_\varrho|^2\,dx&\leq& c\varrho^2\int_{B_\varrho}|\nabla (\dd w)|^2\,dx \\ &\leq& c\varrho^2\int_{B_\varrho}|\nabla (D_\tau w)|^2\,dx \\ &\leq& c\int_{B_{2\varrho}}|D_\tau w-(D_\tau w)_{2\varrho}|^2\,dx, \end{eqnarray*} hence (\ref{zeq4}) yields \begin{equation}\label{zeq44} \int_{B_\varrho}|\dd w-(\dd w)_\varrho|^2\,dx\leq c\biggl(\frac{\varrho}{R}\biggr)^{n+2\gamma} \int_{B_R}|D_\tau w-(D_\tau w)_R|^2\,dx. \end{equation} The same estimates are valid in balls not centered at the origin, hence the isomorphism between the Campanato space ${\cal L}^{2,n+2\gamma}$ and $C^{0,\gamma}$ yields $$ \Vert \dd w\Vert_{C^{0,\gamma}(B_{R/2})}\leq cR^{-n- 2\gamma}\int_{B_R} |\nabla w|^2\,dx.$$ Since the integral of $\dd w$ on $B_{R/2}$ can be estimated with the integral of $\nabla w$ on $B_{R/2}$ we infer \begin{equation}\label{zeq6} R^n\sup_{B_{R/2}} |\dd w|^2\leq c\int_{B_R} |\nabla w|^2\,dx. \end{equation} {\sc Step 3.} Let $\hat{v}(x)=w(x-x_0)$. By (\ref{eq5}) and (\ref{zeq1}) we infer \begin{eqnarray*} &&\int_{B_R(x_0)}\bar{A}_{ij}(x)\left( \frac{\partial v}{\partial x_i}-\frac{\partial \hat{v}}{\partial x_i} \right)\frac{\partial\phi}{\partial x_j}\,dx \\ &=& \int_{B_R(x_0)}\left[\bar{A}_{ij}(x)-A_{ij}(x)\right] \frac{\partial v}{\partial x_i}\frac{\partial\phi}{\partial x_j}\,dx+ \int_{B_R(x_0)} g(x)\phi(x) \,dx \end{eqnarray*} for any $\phi\in H^1_0(B_R(x_0))$. Choosing $\phi=v-\hat{v}$, the ellipticity of $\bar{A}$, (\ref{zeq2}) and Poincar\'e inequality yield \begin{equation}\label{zeq7} \int_{B_R(x_0)}|\nabla v-\nabla\hat{v}|^2\,dx\leq cR^{2\rho} \int_{B_R(x_0)}|\nabla v|^2\,dx+cR^{n+2}. \end{equation} To estimate the Dirichlet integral we notice that (we use (\ref{zeq3}), (\ref{zeq6}) and (\ref{zeq7})) for $0<\varrho\leq R/2$ we have \begin{eqnarray*} &&\int_{B_\varrho(x_0)}|\nabla v|^2\,dx \\ &\leq& 2\int_{B_\varrho(x_0)}|\nabla\hat{v}|^2\,dx+2\int_{B_\varrho(x_0)} |\nabla v-\nabla\hat{v}|^2\,dx \\ &\leq& c\varrho^n\sup_{B_{R/2}(x_0)}|\nabla\hat{v}|^2+ cR^{2\rho}\int_{B_R(x_0)}|\nabla v|^2\,dx+cR^{n+2} \\ &\leq& c\left(\frac{\varrho}{R}\right)^n\int_{B_R(x_0)}|\nabla\hat{v}|^2\,dx+ cR^{2\rho}\int_{B_R(x_0)}|\nabla v|^2\,dx+cR^{n+2} \\ &\leq& c\left(\frac{\varrho}{R}\right)^n\int_{B_R(x_0)}|\nabla v|^2\,dx+ cR^{2\rho}\int_{B_R(x_0)}|\nabla v|^2\,dx+cR^{n+2} \end{eqnarray*} hence, by applying Lemma \ref{Lp} below, with $\alpha=n$, $\beta0$. Then, the supremum of $f(\varrho)/\varrho^\beta$ in $(0,R_0]$ is bounded by a constant depending only on $A,\,B,\,\alpha,\, \beta,\,\rho$, $R_0$ and $f(R_0)$. \end{lemma} {\sc Proof.} Let us fix $\beta<\gamma<\alpha$ and $0<\tau\leq 1/2$ such that $2A\tau^\alpha \leq\tau^\gamma$. If we choose $R_1\leq R_0$ such that $R_1^\rho \leq\tau^\alpha$ from the assumption we get for any $i=1,2,\ldots$ $$ f(\tau^{i+1}R_1)\leq\tau^\gamma f(\tau^iR_1)+B\tau^{i\beta}R_1^{\beta}. $$ Therefore, iterating this estimate, we obtain for any positive integer $k$ \begin{eqnarray*} f(\tau^{k+1}R_1) &\leq& \tau^{\gamma k}f(\tau R_1)+B\tau^{k\beta} R_1^{\beta}\sum_{i=1}^{k-1}\tau^{i(\gamma-\beta)} \\ &\leq& c\tau^{k\beta}\left[ f(R_1)+BR_1\right] . \end{eqnarray*} If $\varrho\leq R_1/2$, choosing $k$ such that $\tau^{k+1}R_1<\varrho \leq\tau^k R_1$, we then get $$ \frac{f(\varrho)}{\varrho^\beta}\leq c\biggl[\frac{f(R_1)}{R_1^\beta}+ B\biggr]\leq c\left[ f(R_0)+B\right] $$ and from this inequality the result immediately follows. \qed \section{Regularity of solutions of the mean curvature equation} In this section we prove an interior regularity result for solutions $\phi\in C^1(D)$ of the prescribed mean curvature equation \begin{equation}\label{mcm1} -{\rm div}\left(\frac{\nabla\phi}{\sqrt{1+|\nabla\phi|^2}}\right)=H \qquad\qquad\hbox{\sl in }D. \end{equation} Here $D\subset\rn{m}$ is an open set; coupling the results of this section (with $m=(n-1)$) with the results of the previous one, the higher regularity of the discontinuity set will be achieved in the forthcoming section. Our result is the following: \begin{theorem}\label{meanreg} Assume that (\ref{mcm1}) holds with $H\in L^\infty(D)$ and $\phi\in C^1(D)$. Then, $\phi\in C^{1,1}_{\rm loc}(D)$ if $m=1$ and $\phi\in H^2_{\rm loc}(D)$, $\nabla\phi\in C^{0,\rho}_{\rm loc}(D;\rn{m})$ for any $\rho<1$ if if $m>1$. \end{theorem} {\sc Proof.} In the case $m=1$, the equation trivially implies that $\phi'/\sqrt{1+|\phi'|^2}$ is locally Lipschitz continuous in $D$, hence $\phi$ is locally $C^{1,1}$ in $D$. In the case $m>1$, we notice that $\phi$ satisfies an equation of the form \begin{equation}\label{eqF} -{\rm div} \bigl( F(\nabla\phi)\bigr)=H\in L^\infty(D) \end{equation} with $F(z)=z/(1+|z|^2)$. Being $F$ the gradient of $\sqrt{1+|z|^2}$, $F$ is locally strictly monotone. Then, the $H^2$ regularity of $\phi$ follows by Poposition~\ref{H2} below; the H\"older continuity of $\nabla\phi$ follows by writing the equation in non divergence form $$ \sum_{i,\,j=1}^m A_{ij}(x){\partial^2\phi\over\partial x_i\partial x_j}(x)=-H(x) $$ with $$ A_{ij}(x)=\frac{\delta_{ij}-\nu_i(x)\nu_j(x)}{(1+|\nabla\phi(x)|^2)^{1/2}}, \qquad \nu_i(x)= \frac{\frac{\partial\phi}{\partial x_i}(x)} {(1+|\nabla\phi(x)|^2)^{1/2}} $$ and applying the regularity theory for equations in non-divergence form with continuous coefficients (see for instance \cite{gt}, Theorem 9.15). \qed \begin{proposition}\label{H2} Let $\phi\in C^1(D)$ be a solution, in the distribution sense, of equation (\ref{eqF}), where $F\in C^1(\rn{m})$ is a locally strictly monotone function, {\em i.e.}, for any compact set $K\subset\rn{m}$ we have \begin{equation}\label{mono} (F(z)-F(z'),z-z')\geq c_K|z-z'|^2\qquad\qquad\forall z,\,z'\in K \end{equation} for a suitable constant $c_K>0$. Then, $\phi\in H^2_{\rm loc}(D)$. \end{proposition} {\sc Proof.} Possibly reducing $D$ we can assume that the gradient of $\phi$ is bounded by a constant $M$ on $D$. We also assume, to simplify some integral estimates, that $D$ is a ball of radius less than 1. We denote by $c$ the constant in (\ref{mono}) corresponding to the compact set $K=\overline{B}_M(0)$ and by $L$ the Lipschitz constant of $F$ on $K$. Given an integer $s$ in $[1,m]$, let $\Delta_hv(x)=[v(x+he_s)-v(x)]/h$ be the difference quotient of a generic function $v$ in the direction $e_s$. We will use the following inequality \begin{equation}\label{basic} \int_{\rn{m}}\psi^2(x)|\Delta_{-h} v(x)|\,dx\leq \int_{\rn{m}}\chi_h(x)|\nabla_s v(x)|\,dx \end{equation} with $$ \chi_h(x)=\int_0^1 \psi^2(x+the_s)\,dt,$$ which is true provided $\psi$ is continuous with compact support, and $v$ is $C^1$ in the $h$-neigbourhood of the support of $\psi$. The proof is standard, being based on the integration of $$ \Delta_{-h}v(x)=\int_0^1 \nabla_s v(x-the_s)\,dt $$ and Fubini's theorem, so we omit it. Let $\eta\in C^1_0(D)$; applying (\ref{eqF}) (for $h$ sufficiently small, depending on the support of $\eta$) with test functions $\eta(x)$ and $\eta(x-he_s)$ and subtracting both sides we get \begin{eqnarray} \label{basic1} &&{1\over h}\int_D [F_i(\nabla\phi(x+he_s))-F_i(\nabla\phi(x))] \nabla_i\eta(x)\,dx \nonumber \\ &=&-\int_D H(x)\Delta_{-h}\eta(x)\,dx. \end{eqnarray} Now we choose $\psi\in C^1_0(D)$ such that $0\leq\psi\leq 1$ and apply (\ref{basic1}) with $\eta=\psi^2\Delta_h\phi$; the integral containing $H$ is equal to $$\int_D H\Delta_{-h}\psi^2\Delta_{-h}\phi\,dx+ \int_D H\psi^2\Delta_{-h}\Delta_h\phi\,dx$$ and can be estimated from above, using (\ref{basic}) with $v=\Delta_h\phi$, by $$ \Vert H\Vert_\infty\bigl[2M\Vert\nabla\psi\Vert_\infty+ \int_D\chi_h|\Delta_h\nabla_s\phi|\,dx\bigr]. $$ Adding and subtracting $\psi^2$, and using $\Vert\chi_h-\psi^2\Vert_\infty\leq 2h\Vert\nabla\psi\Vert_\infty$, this term can be estimated from below by $$ \Vert H\Vert_\infty\bigl[6M\Vert\nabla\psi\Vert_\infty+ \int_D\psi^2|\Delta_h\nabla_s\phi|\,dx\bigr]. $$ Now we estimate from below the first term in (\ref{basic1}). Since $\nabla\eta=\nabla\psi^2\Delta_h\phi+\psi^2\Delta_h\nabla\phi$, using (\ref{mono}) we obtain the estimate $$ c\int_D \psi^2|\Delta_h\nabla\phi|^2\,dx- 2LM\Vert\nabla\psi\Vert_\infty\int_D \psi|\Delta_h\nabla\phi|\,dx.$$ Taking into account these estimates, we obtain that $$ \int_D \psi^2|\Delta_h\nabla\phi|^2\,dx$$ is bounded as $h\to 0$. Hence, $\nabla_s\phi\in H^1$ in the interior of the support of $\psi$. Since $\psi$ is arbitrary, the statement follows.\qed \section{Higher regularity of the discontinuity set} Let $A\subset\rn{n}$ be an open set, $g\in L^\infty(A)$, and let $(u,\Gamma)$ be an admissible pair for the Mumford-Shah functional $F$, minimizing $F$ with respect to relatively compact perturbations in $A$ of $u$ and $\Gamma$. In this section we assume that $\Gamma\cap A$ is a $C^{1,\rho}$ hypersurface, we assume for simplicity $\alpha=\beta=1$ in (\ref{msfunct}) and we investigate the higher regularity of $\Gamma$. There is no loss of generality if we assume that $A=D\times (-1,1)$ for some open set $D\subset\rn{n-1}$ and that $\Gamma\cap A$ is the graph of a $C^{1,\rho}$ function $\phi$ defined on $D$ such that $\Vert\phi\Vert_{\infty}=\varrho <1$. We denote by $A^+\subset A$ and $A^-\subset A$ respectively the epigraph and the hypograph of $\phi$. By the regularity results of the preceding sections (see Theorem~\ref{neureg} or, in the case $n=2$, Theorem~\ref{neureg1}) applied with $\Om=A^\pm$, we can also assume that $u$ and $\nabla u$ are H\"older continuous up to $\Gamma$. We will denote by $(z,y)$ the generic point in $A$, with $z\in D$ and $y\in (-1,1)$. We first see what can be said without extra regularity assumptions on $g$: \begin{theorem}\label{meanreg1} Under the assumptions above, $\phi$ is a solution in the distribution sense of the prescribed mean curvature equation (\ref{mcm1}) for some $H\in L^\infty(D)$. In particular, by Theorem~\ref{meanreg}, $\phi\in C^{1,1}_{\rm loc}(D)$ if $n=2$ and $\phi\in H^2_{\rm loc}(D)$, $\nabla\phi\in C^{0,\rho}_{\rm loc}(D;\rn{n-1})$ for any $\rho<1$ if $n>2$. \end{theorem} {\sc Proof.} We will first show a minimality condition satisfied by $\phi$; then, a suitable choice of the comparison function will allow to conclude. {\sc Step 1.} Let us prove, by a deformation argument, that $\phi$ satisfies the following minimality condition: \begin{equation}\label{qmin} \int_D\sqrt{1+|\nabla\phi|^2}\,dz\leq \int_D\sqrt{1+|\nabla(\phi+\psi)|^2}\,dz+ \lambda\int_D |\psi(z)|\,dz \end{equation} for any function $\psi\in C^1_0(D)$ such that $\Lambda\Vert\psi\Vert_{C^1} \leq 1$. The constants $\lambda$, $\Lambda$ depend only on $\phi$ but not on $\psi$. Indeed, let $\Gamma'$ be the graph of $\phi+\psi$ and assume that $\Lambda$ is so large that $\Lambda\Vert\psi\Vert_{C^1}\leq 1$ implies $\Vert\phi\Vert_\infty+2\Vert\psi\Vert_\infty\leq 1$ and $\Vert\nabla\psi\Vert_\infty\leq 1$. Let $x=(z,y)$ and let $\Phi:A\to A$ be the map defined by $\Phi(z,y)=(z,L_z(y))$ where $L_z(y):[-1,1]\to [-1,1]$ is a piecewise linear bijection, $1/2\leq L_z'\leq 3/2$, and for any $z\in D$ satisfies the properties $$ L_z(\phi(z))=\phi(z)+\psi(z),\qquad L_z(y)=y\quad\hbox{\rm if }|y-\phi(z)|\geq 2|\psi(z)|. $$ It is easy to check that the restriction of $\Phi$ to $\partial A$ is the identity map, and that $\Phi$ is invertible. Moreover, since the coefficients of $L_z(y)$ are linear functions of $\phi(z)$, $\psi(z)$, the Lipschitz constant of $\Phi^{-1}$ can be estimated with the Lipschitz constants of $\phi,\,\phi+\psi$. In addition, \begin{equation}\label{Phiid} {\cal L}^n\bigl(\{x\in A:\Phi(x)\neq x\}\bigr)\leq 4\int_D |\psi(z)|\,dz. \end{equation} The function $v=u\circ\Phi^{-1}$ has the same trace of $u$ on $\partial A$ and its discontinuity set is contained in $\Gamma'$. Taking into account the boundedness of $u$ and $\nabla u$, comparing the energy of $(u,\Gamma)$ with the energy of $(v,\Gamma')$ and taking into account $$ {\cal L}^n\bigl(\{x\in A:u(x)\neq v(x)\}\bigr)\leq 4\int_D |\psi(z)|\,dz. $$ we obtain $$ {\cal H}^{n-1}(\Gamma)\leq {\cal H}^{n-1}(\Gamma')+ 4\lambda\int_D |\psi(z)|\,dz$$ for $\lambda$ large enough (depending on the bounds on $\nabla u$ and on $\nabla\Phi^{-1}$), as claimed. {\sc Step 2.} Let $\varphi\in C^1_0(D)$ and $\eps>0$. Taking $\psi=\eps\varphi$ in the minimality condition (\ref{qmin}), differentiation in $\eps$ gives $$ -\int_D \frac{\langle\nabla\phi,\nabla\varphi\rangle} {\sqrt{1+|\nabla\phi|^2}}\,dz \leq 4\lambda\Vert\varphi\Vert_1.$$ Since $\varphi$ is arbitrary, equation (\ref{mcm1}) is satisfied in the distribution sense for some function $H$ whose $L^\infty$ norm does not exceed $4\lambda$.\qed Now we make additional regularity assumptions on $g$. We need at least $C^1$ regularity to start the bootstrap argument. \begin{theorem}\label{meanreg2} If $g\in C^{k,\rho}_{\rm loc}(A)$ for some integer $k\geq 1$ and some $\rho\in (0,1]$, then there exists $s\in (0,1]$ depending on $n$,\, $\rho$ such that $\phi$ belongs to $C^{k+2,s}_{\rm loc}(D)$. In particular, the smoothness of $g$ implies the smoothness of the free discontinuity set $\Gamma$. \end{theorem} {\sc Proof.} For simplicity's sake, let be $\alpha = \beta = 1$. We first assume $k=1$. We know that $u$ satisfies the equation \begin{equation}\label{neu0} \Delta u=(u-g)\quad\hbox{\rm in }A^\pm,\qquad {\partial u\over\partial \nu}=0\quad\hbox{\rm in }\Gamma. \end{equation} We recall that, by Theorem~\ref{neureg}, $u$ and $\nabla u$ are H\"older continuous up to $\Gamma$, on both sides. Orienting with the inner normal to $A^+$ $$ \nu=\frac{(-\nabla\phi ,1)}{\sqrt{1+|\nabla\phi|^2}} $$ we will prove by a deformation argument that the scalar mean curvature of $\Gamma$ at $x$ is given by the H\"older continuous function \begin{equation}\label{mcm2} H(x)=\bigl[|\nabla u|^2+(u-g)^2\bigr](x), \end{equation} where $[\ \cdot\ ]$ denotes the jump across $\Gamma$. Indeed, if $\eta\in C_0^\infty(A)$ is given, setting for $\eps$ small $\tau_\eps (x)= x+\eps\eta(x)$ and $u_\eps(\tau_\eps(x))=u(x)$, computing the first variation along $u_\eps$ leads to the following equation: \begin{eqnarray} \label{firstvar} &&\int_A\bigl[\bigl(|\nabla u|^2+(u-g)^2\bigr){\rm div}\eta- 2(u-g)\langle\nabla g,\eta\rangle - 2\langle\nabla u,\nabla u\nabla\eta\rangle\bigr] dx \nonumber \\ &&=-\int_\Gamma\delta_i\eta_i d{\cal H}^{n-1} \end{eqnarray} where $\delta$ stands for the tangencial differential operator. Fix a cut-off function $\chi\in C_0^1(-1,1)$ such that $\chi\equiv 1$ in $[-\varrho,\varrho]$ (recall that $\varrho=\Vert\phi\Vert_\infty$) and define $\eta_i(x)=0$ for $i1$, the higher regularity of $u$ up to $\Gamma$ and of $\phi$ follows by the usual bootstrap argument, based on (\ref{neu0}), (\ref{mcm1}) and (\ref{mcm2}). \qed \section{Density of functions with regular jump set} In this section we show that in Mumford-Shah problem the Lavrentiev phenomenon does not occur, {\em i.e.}, that the infimum of the functional $F$ among all pairs $(\Gamma,u)$ with $\Gamma$ piecewise smooth (according to Definition~\ref{piesmo}) and $u\in C^\infty(\Omega\setminus\Gamma)$ coincides with the minimum of the weak form of $F$. This property is particularly useful in connection with the numerical minimization of the Mumford-Shah functional (see for instance \cite{cham}, \cite{belco}). Our strategy is to prove a more general approximation theorem of $SBV$ functions by smooth functions with discontinuities contained in piecewise smooth sets. A byproduct of this theorem is the absence of the Lavrentiev phenomenon for general classes of functionals, including the Mumford- Shah functional (see Remark~\ref{Lav}). The proof of the approximation property is achieved in two steps: first, using the partial regularity results of \cite{ap}, \cite{afp} and section 4, we prove in Proposition~\ref{appmin} that the approximation property holds for minimizers of the Mumford- Shah functional. Successively we see that a more general class of $SBV$ functions can be approximated by minimizers. Let us start by recalling the definition and the main properties of the $SBV$ functions; for more details we refer to \cite{dga}, \cite{am3}, \cite{am5}, \cite{am2}. A function $u\in L^1(\Om)$ is said to be in $SBV(\Om)$ if its distributional gradient $Du$ is a Radon vector measure with finite total variation ({\em i.e.} $u\in BV(\Om)$) and moreover $Du=\nabla u\, dx+Ju$, where $Ju$ is the restriction of $Du$ to the complement $S_u$ of the Lebesgue set of $u$ and $Ju$ is absolutely continuous with respect to ${\cal H}^{n-1}\res S_u$. By $\nabla u$ we denote the density of $Du$ with respect to the Lebesgue measure; it can be interpreted as an approximate differential of $u$. The weak form of the Mumford-Shah functional in $SBV(\Om)$ can then be formulated as follows: $$ G(u)=\int_\Om\bigl[|\nabla u|^2+\alpha|u-g|^2\bigr]\, dx + \beta{\cal H}^{n-1}(S_u) $$ and it can be proved that a minimizer $u\in SBV(\Om)$ exists (see \cite{am2}). Let us now define the class of piecewise smooth manifolds that will be considered. \begin{definition}\label{piesmo} Let $K\subset\rn{n}$ be compact. We say that $K$ is piecewise smooth if there exist finitely many $C^\infty$ hypersurfaces $\Gamma_1,\ldots,\Gamma_N$ with $C^\infty$ boundaries $\partial\Gamma_1$, ..., $\partial\Gamma_N$ such that $K=\bigcup\limits_{i=1}^N\Gamma_i$ and $$ \bigl(\Gamma_i\setminus\partial\Gamma_i\bigr)\cap \bigl(\Gamma_j\setminus\partial\Gamma_j\bigr)=\emptyset $$ whenever $i\neq j$. We denote by ${\cal K}$ the class of these sets. \end{definition} The main result of this section is the following theorem, which is an immediate consequence of Theorem \ref{lavre}. \begin{theorem} \label{lavrent} Let $\Omega\subset\rn{n}$ be a bounded open set with Lipschitz boundary, $\alpha,\,\beta$ positive and $g\in L^\infty(\Omega)$; then $$ \inf\bigl\{ F(u,K):\ K\in {\cal K},\ u\in C^\infty(\Omega\setminus K) \Bigr\} = \min\bigl\{ G(u):\ u\in SBV(\Omega) \Bigr\} , $$ where $F$ is defined in (\ref{msfunct}). \end{theorem} \begin{proposition}\label{appmin} Let $\Om,\,A\subset\rn{n}$ be bounded open sets with $\Om\subset\subset A$, let $w\in C^\infty(A)$ and let $(v,\Gamma)$ be a minimizer of the Mumford-Shah functional $$ (v,\Gamma)\mapsto \int_A\bigl[|\nabla v|^2+\alpha |v-w|^2\bigr]dx + \beta {\cal H}^{n-1} (\Gamma\cap A). $$ Then, for every $\eps>0$ we can find a compact set $K\in{\cal K}$ and $u\in C^\infty(\overline{\Omega}\setminus K)$ such that \begin{equation}\label{require} \left\{\begin{array}{l} \displaystyle{\int_\Omega \bigl[ \alpha|v-u|^2+|\nabla v-\nabla u|^2\bigr]\,dx<\eps} \\ \\ \displaystyle{\beta\bigl\vert{\cal H}^{n-1}(K\cap\overline{\Omega})- {\cal H}^{n-1}(\Gamma\cap\overline{\Omega}) \bigr\vert<\eps.} \end{array}\right. \end{equation} \end{proposition} {\sc Proof.} Let $S\subset\Gamma$ be the singular set of $\Gamma$, {\em i.e.}, the set of points $x\in\Gamma$ such that $\Gamma$ is not a $C^{1,\rho}$ hypersurface in any neighbourhood of $x$. We know from Theorem~3.1 and Remark~3.2 of \cite{afp} that ${\cal H}^{n-1}(S)=0$; moreover, Theorem~\ref{meanreg2} implies that $\Gamma\setminus S$ is locally a $C^\infty$ hypersurface. For any $\delta\in (0,1)$ we can find a finite family of balls $B_{r_i}(x_i)$ centered at points $x_i\in\overline{\Om}$ whose union contains $S\cap\overline{\Om}$, such that $r_i<\delta$ for any $i=1,\ldots,N$ and $$ \sum_{i=1}^N r_i^{n-1}<\delta.$$ Let $\chi\in C^\infty_0(A)$ such that $\chi\equiv 1$ on the $\delta$ neighbourhood of $\overline{\Om}$. Denoting by $U$ the union of the balls $B_{\tau r_i}(x_i)$, we define the function $$ u(x)=\cases{ v(x) & if $x\notin U$ \cr 0 & if $x\in U$},$$ the compact set $$ K=\bigl(\Gamma\setminus (U\cup \{ 1+\chi\geq\tau \})\bigr) \cup\bigcup_{i=1}^N \partial B_{\tau r_i}(x_i) $$ and we check that for a suitable choice of $\tau\in (1,2)$ and for $\delta$ sufficiently small, $(u,K)$ satisfy all the stated properties. Indeed, since $r_i<\delta<1$, we have $$ {\cal L}^n(U)\leq \sum_{i=1}^N \omega_n(2r_i)^n<2^n\omega_n\delta, $$ so that $$ \int_\Om \bigl[\alpha |v-u|^2+|\nabla v-\nabla u|^2\bigr]\,dx= \int_U |v|^2+|\nabla v|^2\,dx<\eps $$ for $\delta$ small enough. Since the measure of the symmetric difference of $K$ and $\Gamma$ in $\overline{\Om}$ does not exceed the sum of the perimeter of the balls, also the second condition in (\ref{require}) is fulfilled for $\delta$ small enough. Finally, by Sard's theorem, the balls $B_{\tau r_i}(x_i)$ and the smooth level set $\{\chi=\tau-1\}$ meet transversally $\Gamma$ for a.e. $\tau$, hence a choice of $\tau\in (1,2)$ is possible ensuring that $K\in {\cal K}$. \qed \begin{theorem}\label{lavre} Let $\Omega\subset\rn{n}$ be a bounded open set with Lipschitz boundary. Then, for every $u\in SBV(\Omega)\cap L^\infty(\Omega)$ satisfying $$\int_\Omega |\nabla u|^2\,dx+{\cal H}^{n-1}(S_u)<+\infty$$ we can find sets $K_h\in {\cal K}$ and functions $u_h\in C^\infty(\overline{\Omega}\setminus K_h)$ such that $$\left\{ \begin{array}{l} \displaystyle{\lim\limits_{h\to +\infty}\int_\Omega\bigl[ |u_h- u|^2+|\nabla u_h-\nabla u|^2 \bigr]\,dx=0}, \\ \\ \displaystyle{\lim\limits_{h\to +\infty}{\cal H}^{n- 1}(K_h\cap\overline{\Omega})= {\cal H}^{n-1}(S_u\cap\Omega).} \end{array}\right.$$ \end{theorem} {\sc Proof.} As a first step, we consider a bounded open set $A\subset\rn{n}$ such that $\overline{\Omega}\subset A$ and a function $w\in SBV(A)$ such that $w=u$ on $\Omega$, ${\cal H}^{n-1}(S_w\cap\partial\Omega)=0$ and $$\int_A |\nabla w|^2\,dx+{\cal H}^{n-1}(S_w\cap A)<+\infty.$$ An extension with these properties can be easily obtained if $\Omega$ is a rectangle, by a reflection argument. In the general case the extension can be obtained using partitions of unity and a deformation argument. Let $(w_h)\subset C^\infty(A)$ be such that $h\Vert w_h- w\Vert_{L^2(A)}\to 0$ as $h\to +\infty$, and let $v_h\in SBV(A)$ be minimizers of the functionals $$ v\mapsto {\cal F}_h(v)= \int_A \bigl[|\nabla v|^2+h(v-w_h)^2\bigr]\,dx+ {\cal H}^{n-1}(S_v\cap A).$$ Since ${\cal F}_h(v_h)\leq {\cal F}_h(w)$, we obtain \begin{eqnarray}\label{stima1} &&\limsup_{h\to +\infty} \biggl[h\Vert v_h-w_h\Vert^2_{L^2(A)}+ \int_A |\nabla v_h|^2\,dx+ {\cal H}^{n-1}(S_{v_h}\cap A)\biggr] \\ &\leq& \int_A |\nabla w|^2\,dx+{\cal H}^{n-1}(S_w\cap A).\nonumber \end{eqnarray} In particular, $\Vert v_h-w_h\Vert_{L^2(A)}\to 0$, hence $v_h$ converges to $w$ in $L^2(A)$. Moreover, the $SBV$ compactness theorem (see \cite{am1}, \cite{am4}) implies $$\left\{\begin{array}{l} \liminf\limits_{h\to +\infty}\int_A|\nabla v_h|^2\,dx\geq \int_A |\nabla w|^2\,dx \\ \liminf\limits_{h\to +\infty}{\cal H}^{n-1}(S_{v_h}\cap A)\geq {\cal H}^{n-1}(S_w\cap A). \end{array} \right. $$ Then, these inequalities and (\ref{stima1}) imply $$\left\{\begin{array}{l} \lim\limits_{h\to +\infty}\int_A |\nabla v_h|^2\,dx=\int_A |\nabla w|^2\,dx \\ \lim\limits_{h\to +\infty}{\cal H}^{n-1}(S_{v_h}\cap A)= {\cal H}^{n-1}(S_w\cap A). \end{array} \right. $$ In particular, $\nabla v_h$ converges strongly to $\nabla w$ as $h\to +\infty$ and ${\cal H}^{n-1}(S_{v_h}\cap A)$ converges to ${\cal H}^{n-1}(S_w\cap A)$. Again, the $SBV$ compactness theorem gives $$\left\{\begin{array}{l} \liminf\limits_{h\to +\infty}{\cal H}^{n-1}(S_{v_h}\cap\Omega)\geq {\cal H}^{n-1}(S_w\cap\Omega) \\ \liminf\limits_{h\to +\infty}{\cal H}^{n-1}\bigl(S_{v_h}\cap(A\setminus \overline{\Omega})\bigr)\geq {\cal H}^{n-1}\bigl(S_w\cap(A\setminus\overline{\Omega})\bigr). \end{array} \right. $$ Since ${\cal H}^{n-1}(S_w\cap\partial\Omega)=0$, these inequalities imply that both liminf above are limits and that equalities hold. In particular, ${\cal H}^{n-1}(S_{v_h}\cap\Omega)$ converges to ${\cal H}^{n-1}(S_u\cap\Omega)$ and \begin{equation}\label{bordo} \lim_{h\to +\infty}{\cal H}^{n-1}(S_{v_h}\cap\partial\Omega )=0. \end{equation} Let $\Gamma_h=\overline{S}_{v_h}$. By the minimality of $v_h$ and the smoothness of $w_h$, the functions $v_h$ are $C^\infty$ in $A\setminus\Gamma_h$; moreover, (\ref{bordo}) and (see \cite{ap}, Proposition 2.8) $$ {\cal H}^{n-1}(A\cap\Gamma_h\setminus S_{v_h})=0$$ imply that ${\cal H}^{n-1}(\Gamma_h\cap\overline{\Omega})$ converges to ${\cal H}^{n-1}(S_u\cap\Omega)$. However, the sets $\Gamma_h$ do not necessarily belong to ${\cal K}$; using Proposition~\ref{appmin} we can modify $(v_h,\Gamma_h)$ to obtain pairs $(u_h,K_h)$ satisfying the statement.\qed \begin{remark} \label{Lav} {\rm As a consequence of the approximation Theorem~\ref{lavre}, we obtain that the Lavrentiev phenomenon does not occur for minimum problems in $SBV$ of the form $$\min\biggl\{\int_\Omega f(x,u,\nabla u)\,dx+{\cal H}^{n- 1}(S_u)\biggr\}$$ where $f$ is any Carath\'eodory function satisfying quadratic growth conditions. 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